# Questions tagged [groupoids]

A groupoid is a small category in which every morphism is an isomorphism. They arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag. Also, use other tags such as monoid or category-theory, if needed.

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### The small concrete category $C = (\mathbb{Z_2}, +)$?

I'm grappling with the question of viewing a group as a category, which as I understand it means that if $(G,+)$ is the group in question, then the group can also be thought of as a small concrete ...
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### Notation for the scope of definition of a partial binary operation of a groupoid

I have a groupoid $(G,\ast )$ with a partial binary operation $*:G\times G\rightharpoonup G$. For every $(a,b)\displaystyle ∈G\times G$, $\displaystyle *$ is defined if ...
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### Name for semigroupoid-like structure applicable to a water flow network in graph theory

Hope you are well. I am studying flow networks (flows of water) in graph theory. By flow network I mean a graph $G = (V,E)$ where $V$ is a set of $ℝ_{>0}$ labelled vertices (water tanks filled with ...
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### Group isomorphism between identity morphisms of fundamental groupoid and fundamental group of topological space

For any topological space $X$, I have defined $C=\Pi_1(X)$ by objects $\text{Ob}(C)=X$ and morphisms $\text{Hom}_C(x,y)=\text{HPath}(x,y)$, where $\text{HPath}(x,y)$ denotes the set of homotopy ...
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### Homotopy cardinality of weak quotients

Let $G$ be a group (regarded as a one-object groupoid) acting on a groupoid $X$ (i.e.: a functor $F: G \to \mathsf{Groupoids}$ sending the unique object of $G$ to $X$). Denote with $X//G$ the “weak ...
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### Definition of Haar systems, especially for groupoids $C^*$-algebra

In A Groupoid approach to $C^*$-algebras Jean Renault introduces left Haar systems. Say $G$ is a groupoid. Let $\Lambda=\{\mu^u,\,\,:\,\,u\in G^0\}$ be a family of Haar measures on $G$. Renault ...
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### Help with understanding the definition of a Hodge groupoid

I am reading this PhD thesis and I can't understand definition 6.21: (Hodge groupoid). $(X/S)_{Hod}$ is a groupoid whose object object is $X\times \Bbb A^1_S$ and whose morphism object $N_{Hod}$ is ...
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### How units of a Lie groupoid act on a manifold?

The definition of a Lie groupoid action given in Sébastien Racanière's notes (up to notation) says that the action of a Lie groupoid $\mathcal{G} \rightrightarrows M$ on a smooth manifold $Q$ consists ...
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### Cancellation law in a Lie groupoid

I'm playing around with the definition of a Lie groupoid following Eckhard Meinrenken's notes. I read the thing for the first time in my life like an hour ago, so assume that I don't know anything ...
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Sorry for the block of text from Kirill Mackenzie's chapter on groupoids. I have proven that $\natural,\natural_0$ is a groupoid morphism and that this quotient construction satisfies the groupoid ...